The ancient Greeks present: Rational Trigonometry

TL;DR

This paper introduces a simplified and elegant form of trigonometry based on quadrance and spread, extending classical Greek results to more general mathematical contexts.

Contribution

It presents a new formulation of trigonometry using quadrance and spread, generalizing ancient Greek geometric results to arbitrary fields and quadratic forms.

Findings

Simplifies classical theorems using quadrance and spread

Extends trigonometry to arbitrary fields and quadratic forms

Provides a more elegant framework for geometric relationships

Abstract

Pythagoras' theorem, the area of a triangle as one half the base times the height, and Heron's formula are amongst the most important and useful results of ancient Greek geometry. Here we look at all three in a new and improved light, using quadrance not distance. This leads to a simpler and more elegant trigonometry, in which angle is replaced by spread, and which extends to arbitrary fields and more general quadratic forms.